Properties

Label 7.12.8.3
Base \(\Q_{7}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_{12}$ (as 12T1)

Related objects

Learn more about

Defining polynomial

\( x^{12} + 14 x^{9} + 539 x^{6} + 343 x^{3} + 60025 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $12$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.3.2.3, 7.4.0.1, 7.6.4.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:7.4.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{4} + x^{2} - 3 x + 5 \)
Relative Eisenstein polynomial:$ x^{3} - 7 t^{2} \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{12} - x^{11} + x^{10} - 27 x^{9} + 27 x^{8} - 183 x^{7} + 326 x^{6} + 649 x^{5} + 131 x^{4} - 573 x^{3} + 1782 x^{2} - 2133 x + 4941$